Geometry & Topology

Quasigeodesic flows and sphere-filling curves

Steven Frankel

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Given a closed hyperbolic 3–manifold M with a quasigeodesic flow, we construct a π1–equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on 3 has a natural compactification to a closed disc that inherits a π1–action. The embedding P3 extends continuously to the compactification, and restricts to a surjective π1–equivariant map P 3 on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic 3–manifolds.

Article information

Geom. Topol., Volume 19, Number 3 (2015), 1249-1262.

Received: 22 July 2013
Revised: 25 February 2014
Accepted: 26 July 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M60: Group actions in low dimensions
Secondary: 57M50: Geometric structures on low-dimensional manifolds 37C27: Periodic orbits of vector fields and flows

quasigeodesic flows Cannon–Thurston pseudo-Anosov flows


Frankel, Steven. Quasigeodesic flows and sphere-filling curves. Geom. Topol. 19 (2015), no. 3, 1249--1262. doi:10.2140/gt.2015.19.1249.

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